IB Math Studies – Functions

Questions and Answers

IB Math Studies Functions

 

1. IB Mathematical Studies SL – Functions, simplest form

How can I find in simplest form f(-x),  f(x-3), f(2x+1) ,where f(x)=2x^2-3x.

Solution

IB Mathematical Studies SL – Functions, simplest form

The functions in simplest form are:

f(-x)= 2(-x)^2-3(-x)=2x^2+3x

 f(x-3)= 2(x-3)^2-3(x-3)=2(x^2-6x+9)-3x+9=2x^2-12x+18-3x+9=2x^2-15x+27

 f(2x+1)= 2(2x+1)^2-3(2x+1)=2((2x)^2+4x+1)-6x-3=

= 8x^2+8x+2-6x-3=8x^2+2x-1

 

2. IB Mathematical Studies SL – Linear Functions

How can we find a formula that describes the cost of running a car if there is a standard cost of $100 plus $0.1 per one kilometer thereafter?

Solution

IB Mathematical Studies SL – Linear Functions

Let x be the kilometers then the linear function that describes the above word problem is:

f(x)= 100+0.1x

 

3. IB Mathematical Studies SL – Quadratic Functions

The profit of a PC manufacturer company of making x personal computers (PC) per day is given by the following formula:

f(x)=-0.2x^2+12x+220

How can we find the profit in dollars in the following cases:

for 10 PCs, 40PCs and 100PCs

Solution

IB Mathematical Studies SL – Quadratic Functions

The profit of the company when being produced 10 PCs is given by

 f(10)=-0.1(10^2)+12(10)+220=-10+120+220=$330

The profit of the company when being produced 40 PCs is given

 f(40)=-0.1(40^2)+12(40)+220=-160+480+220=$540

The profit of the company when it  produces 100 PCs is given

 f(40)=-0.1(100^2)+12(100)+220=-1000+1200+220=$420

4. IB Mathematical Studies SL – Complete the square

How can we complete the square and find the vertex of the following quadratic function:

f(x)=3x^2-9x+15

Solution

IB Mathematical Studies SL – Complete the square

Completing the square is a procedure for reformatting quadratic functions.

Quadratic Functions of the form f(x)=ax^2+bx+c can be rewritten as

f(x)=a(x-h)^2+k

The major reason for doing this is that it allows us to easily see the vertex of the curve f(x)=ax^2+bx+c which is located at the point (h,k).

The general method of completing the square can be shown like this:

f(x)=ax^2+bx+c=a(x^2+\frac{b}{a}x)+c=

  =a((x+\frac{b}{2a})^2-(\frac{b}{2a})^2)+c=

 =a(x-(-\frac{b}{2a}))^2-\frac{b^2}{4a}+c

Therefore the x-coordinate of the vertex is h=-\frac{b}{2a}

Concerning this question if we follow the preceding procedure we have:

f(x)=3x^2-9x+15=3(x^2+\frac{-9}{3}x)+15=

  =3((x+\frac{-9}{2(3)})^2-(\frac{-9}{2(3)})^2)+15=

 =3(x-(-\frac{-9}{2(3)}))^2-\frac{(-9)^2}{12}+15=

 =3(x-(\frac{3}{2}))^2-\frac{81}{12}+15=

 =3(x-(\frac{3}{2}))^2-\frac{81}{12}+\frac{15(12)}{12}=

 =3(x-(\frac{3}{2}))^2-\frac{81}{12}+\frac{180}{12}=

 =3(x-(\frac{3}{2}))^2+\frac{99}{12}=

Therefore the coordinates of the vertex are ( \frac{3}{2}, \frac{99}{12})

 

5. IB Mathematical Studies SL – Factorisation

How can we factorize this expression 4x^2-16?

Solution

IB Mathematical Studies SL – Factorization, difference of squares

We‘ll use the difference of squares identity

a^2-b^2=(a-b)(a+b)

Concerning this question the factorization is

4x^2-16=(2x)^2-4^2=(2x-4)( 2x+4)

 

6. IB Mathematical Studies SL – Factorisation, Common Factor

How can we fully factorize the following expression:

(3x-4)^2-9(3x-4)?

Solution

IB Mathematical Studies SL – Factorisation, common factor

The expression can be factorized as follows

 (3x-4)^2-9(3x-4) =(3x-4)((3x-4)-9)= (3x-4)(3x-13) ?